Bishop uses the villanelle form where the first and third lines of the first tercet are repeated (and sometimes rephrased) in the third line of each subsequent tercet. In the last stanza, a quatrain, these two lines are repeated together as the final two lines. The repetition underscores the frequent occurrences of loss. We lose trivial things often; thus, it isn't hard to master.
Structurally, the poem begins with these trivial losses (keys, an hour, names) and progresses to more significant things and people. From keys to her mother's watch to an entire content, and finally a person, the speaker gets progressively more profound and less flippant about loss. Losing keys is easy to master but losing those more significant things is not. The speaker repeats the line, paraphrased each time, that losing is "no disaster." However, by the end of the poem, she notes that losing is easy to master, although it does at this point look like disaster:
the art of losing's not too hard to master
though it may look like (Write it!) disaster.
At the beginning of the poem, loss was simply not disaster. But in the end, more profound loss looks like disaster. And if it looks like disaster, maybe it is.
The speaker encourages the "you" of the poem ( the reader and in the last stanza, presumably the "you" is her lover who had committed suicide) to practice losing. The speaker casually encourages one to practice losing things like keys since none of these losses will be disastrous. But in the fourth stanza, she stops encouraging and begins describe more personal losses. This is a transition in the poem. Although she continues to repeat that losing isn't hard to master, the tone has shifted.
She describes the "lovely" cities she's lost, her mother's watch, and "three loved houses." She even notes that she misses the rivers and the continent she has lost. Finally, she describes losing someone, the most personal moment, in the last stanza. In the last line, she writes with apostrophe "(Write it!)" as if urging herself to repeat the final phrase that such loss only looks like disaster. In writing "write it," it seems like the speaker is trying to convince herself to accept the loss just like she's accepted losing more trivial things. From this, it is clear that losing profound things is not so easy to master.
There is also the interpretation that there is a pun on losing as "art." As losing is an "art," dealing with loss is also an art. More specifically, expression (writing poetry) is a way of dealing with that loss. So, a poet can use an art to deal with another art (of losing). Therefore, she must "write it" or write about loss as a way of expressing and coming to terms with it.